Expressions Using Letter-Numbers

Write formulas for the perimeter of: (a) triangle with all sides equal. (b) a regular pentagon (as we have learnt last year, we use the word 'regular' to say that all sidelengths and angle measures are equal) (c) a regular hexagon

Munirathna has a 20 m long pipe. However, he wants a longer watering pipe for his garden. He joins another pipe of some length to this one. Give the expression for the combined length of the pipe. Use the letter-number 'k' to denote the length in meters of the other pipe.

What is the total amount Krithika has, if she has the following numbers of notes of ₹100, ₹200 and ₹5? Complete the following table:

Venkatalakshmi owns a flour mill. It takes 10 seconds for the roller mill to start running. Once it is running, each kg of grain takes 8 seconds to grind into powder. Which of the expressions below describes the time taken to complete grind $y$ kg of grain, assuming the machine is off initially?

(a) $10 + 8 + y$

(b) $(10 + 8) \times y$

(c) $10 \times 8 \times y$

(d) $10 + 8 \times y$

(e) $10 \times y + 8$

Write algebraic expressions using letters of your choice.

(a) 5 more than a number

(b) 4 less than a number

(c) 2 less than 13 times a number

(d) 13 less than 2 times a number

Describe situations corresponding to the following algebraic expressions:

(a) $8 \times x + 3 \times y$

(b) $15 \times j - 2 \times k$

In a calendar month, if any $2 \times 3$ grid full of dates is chosen as shown in the picture, write expressions for the dates in the blank cells if the bottom middle cell has date 'w'.

Add the numbers in each picture below. Write their corresponding expressions and simplify them. Try adding the numbers in each picture in a couple different ways and see that you get the same thing.

Simplify each of the following expressions:

(a) $p + p + p + p$, $p + p + p + q$,

(b) $p + q + p - q$,

(c) $p - q + p - q$,

(d) $p + q - p + q$,

(e) $p + q - (p + q)$,

(f) $p - q - p - q$,

(g) $2d - d - d - d$,

(h) $2d - d - d - c$,

(i) $2d - d - (d - c)$,

(j) $2d - (d - d) - c$,

(k) $2d - d - c - c$

One plate of Jowar roti costs ₹30 and one plate of Pulao costs ₹20. If $x$ plates of Jowar roti and $y$ plates of pulao were ordered in a day, which expression(s) describe the total amount in rupees earned that day?

(a) $30x + 20y$

(b) $(30 + 20) \times (x + y)$

(c) $20x + 30y$

(d) $(30 + 20) \times x + y$

(e) $30x - 20y$

Pushpita sells two types of flowers on Independence day: champak and marigold. ‘p’ customers only bought champak, ‘q’ customers only bought marigold, and ‘r’ customers bought both. On the same day, she gave away a tiny national flag to every customer. How many flags did she give away that day?

(a) p + q + r

(b) p + q + 2r

(c) 2 × (p + q + r)

(d) p + q + r + 2

(e) p + q + r + 1

(f) 2 × (p + q)

A snail is trying to climb along the wall of a deep well. During the day it climbs up ‘u’ cm and during the night it slowly slips down ‘d’ cm. This happens for 10 days and 10 nights.

(a) Write an expression describing how far away the snail is from its starting position.

(b) What can we say about the snail’s movement if d > u?

Radha is preparing for a cycling race and practices daily. The first week she cycles 5 km every day. Every week she increases the daily distance cycled by ‘z’ km. How many kilometers would Radha have cycled after 3 weeks?

In the following figure, observe how the expression w + 2 becomes 4w + 20 along one path. Fill in the blanks on the remaining paths. The ovals contain expressions and the boxes contain operations.

A local train from Yahapur to Vahapur stops at three stations at equal distances along the way. The time taken in minutes to travel from one station to the next station is the same and is denoted by $t$. The train stops for 2 minutes at each of the three stations.

(a) If $t = 4$, what is the time taken to travel from Yahapur to Vahapur?

(b) What is the algebraic expression for the time taken to travel from Yahapur to Vahapur? [Hint: Draw a rough diagram to visualise the situation]

Simplify the following expressions:

(a) $3a + 9b - 6 + 8a - 4b - 7a + 16$

(b) $3(3a - 3b) - 8a - 4b - 16$

(c) $2(2x - 3) + 8x + 12$

(d) $8x - (2x - 3) + 12$

(e) $8h - (5 + 7h) + 9$

(f) $23 + 4(6m - 3n) - 8n - 3m - 18$

Add the expressions given below:

(a) $4d - 7c + 9$ and $8c - 11 + 9d$

(b) $-6f + 19 - 8s$ and $-23 + 13f + 12s$

(c) $8d - 14c + 9$ and $16c - (11 + 9d)$

(d) $6f - 20 + 8s$ and $23 - 13f - 12s$

(e) $13m - 12n$ and $12n - 13m$

(f) $-26m + 24n$ and $26m - 24n$

Subtract the expressions given below:

(a) $9a - 6b + 14$ from $6a + 9b - 18$

(b) $-15x + 13 - 9y$ from $7y - 10 + 3x$

(c) $17g + 9 - 7h$ from $11 - 10g + 3h$

(d) $9a - 6b + 14$ from $6a - (9b + 18)$

(e) $10x + 2 + 10y$ from $-3y + 8 - 3x$

(f) $8g + 4h - 10$ from $7h - 8g + 20$

Describe situations corresponding to the following algebraic expressions:

(a) $8x + 3y$

(b) $15x - 2x$

Imagine a straight rope. If it is cut once as shown in the picture, we get 2 pieces. If the rope is folded once and then cut as shown, we get 3 pieces. Observe the pattern and find the number of pieces if the rope is folded 10 times and cut. What is the expression for the number of pieces when the rope is folded $r$ times and cut?

Look at the matchstick pattern below. Observe and identify the pattern. How many matchsticks are required to make 10 such squares. How many are required to make w squares?

Have you noticed how the colours change in a traffic signal? The sequence of colour changes is shown below. Find the colour at positions 90, 190, and 343. Write expressions to describe the positions for each colour.

Observe the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Write a general formula. How would the formula change if we want to count the number of vertices of all the squares?

Context: Let $a$ denote Aftab's age and $s$ denote Shabnam's age. The algebraic expression to find Aftab's age is $a = s - 3$.

Q. Use this expression to find Aftab's age if Shabnam's age is 20.

How much should she pay if she buys 8 coconuts and 9 kg jaggery?

Use this expression (or formula) to find the total amount to be paid for 7 coconuts and 4 kg jaggery.

What is the perimeter of a square with sidelength 7 cm? Use the expression to find out.

Now, find the values of the other arithmetic expressions:

1. $23 - 10 \times 2$
2. $83 + 28 - 13 + 32$
3. $34 - 14 + 20$
4. $42 + 15 - (8 - 7)$
5. $68 - (18 + 13)$
6. $7 \times 4 + 9 \times 6$
7. $20 + 8 \times (16 - 6)$

Context: Consider the number sequence: 4, 8, 12, 16, 20, 24, 28, ...

Q. Find an algebraic expression to get the nth term of this sequence.

Mind the Mistake, Mend the Mistake

Some simplifications are shown below where the letter-numbers are replaced by numbers and the value of the expression is obtained.

1. Observe each of them and identify if there is a mistake.
2. If you think there is a mistake, try to explain what might have gone wrong.
3. Then, correct it and give the value of the expression.

Context: Example 5: Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is $c$, and the price per eraser is $d$.

The total money earned by the sale of pencils is $5c + 3c + 10c = 18c$.

Q. If $c = ₹50$, find the total amount earned by the sale of pencils.

Context: Example 5: Here is a table showing the number of pencils and erasers sold in a shop. The price per pencil is $c$, and the price per eraser is $d$. Find the total money earned by the shopkeeper during these three days.

Q. Write the expression for the total money earned by selling erasers. Then, simplify the expression.

Context: In this problem, we saw the expression $5c + 3c + 10c$ getting simplified to the expression $18c$.

Q. Check that both expressions take the same value when $c$ is replaced by different numbers.

Context: The expression $(40x + 75y) - (6x + 10y)$ is simplified to $34x + 65y$, which is the total amount paid in rupees.

Q. Could we have written the initial expression as $(40x + 75y) + (-6x - 10y)$?

Context: The total amount in rupees paid at the beginning is $40x + 75y$, and the total amount returned is $6x + 10y$. So, the total amount paid = $(40x + 75y) - (6x + 10y)$.

Q. Can we simplify this expression? If yes, how? If not, why not?

What if there is no penalty? What will be the value of $q$ in that situation?

Context: In a quiz, $p$ represents the score for a correct answer and $q$ represents the penalty for an incorrect answer. Krishita's total score after three rounds is $23p - 7q$.

Q. Give some possible scores for Krishita in the three rounds so that they add up to give $23p - 7q$.

Context: Charu's total score after three rounds is $21p - 9q$, and Krishita's total score is $23p - 7q$, where $p$ represents the score for a correct answer and $q$ represents the penalty for an incorrect answer.

Q. Can we say who scored more? Can you explain why?

Context: Consider the expression representing the difference between Krishita's and Charu's scores: $(23p - 7q) - (21p - 9q)$.

Q. Simplify this expression further.

Fill the blanks below by replacing the letter-numbers by numbers; an example is shown. Then compare the values that $5u$ and $5 + u$ take.

Context: Let us compare the values that the expressions $10y - 3$ and $10(y - 3)$ take for different values of $y$.

Q. After filling in the two diagrams, do you think the two expressions are equal?

Mind the Mistake, Mend the Mistake

Some simplifications of algebraic expressions are done below. The expression on the right-hand side should be in its simplest form.

• Observe each of them and see if there is a mistake.
• If you think there is a mistake, try to explain what might have gone wrong.
• Then, simplify it correctly.

Take a look at all the corrected simplest forms (i.e. brackets are removed, like terms are added, and terms with only numbers are also added). Is there any relation between the number of terms and the number of letter-numbers these expressions have?

Find the formulas of the number machines below and write the expression for each set of inputs.

Now, make a formula on your own. Write a few number machines as examples using that formula. Challenge your classmates to figure it out!

Context: Design C appears at positions that are multiples of 3 (remainder 0). Design B appears at positions that are 1 less than a multiple of 3 (remainder 2). Design A appears at positions that are 2 less than a multiple of 3 (remainder 1).

(a) Can the remainder obtained by dividing the position number by 3 be used for this? Observe the table below.

b) Use this to find what design appears at positions 99, 122, and 148.

Context: Let us take the marked $2 \times 2$ square, and consider the numbers lying on the diagonals; 12 and 20; 13 and 19. Find their sums; $12 + 20$, $13 + 19$. What do you observe? They are equal. Let us extend the numbers in the calendar beyond 30, creating endless rows.

Q. Will the diagonal sums be equal in every $2 \times 2$ square in this endless grid? How can we be sure?

Context: Let us consider a $2 \times 2$ square. Its top left number can be any number. Let us call it '$a$'.

Q. Given that we know the top left number, how do we find the other numbers in this $2 \times 2$ square?

Verify this expression for diagonal sums by considering any $2 \times 2$ square and taking its top left number to be 'a'.

Context: Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

Q. Find the sum of all the numbers. Compare it with the number in the centre: 15. Repeat this for another set of numbers that forms this shape. What do you observe?

Q. Will this always happen? How do you show this?

[Hint: Consider a general set of numbers that forms this shape. Take the number at the centre to be 'a'. Express the other numbers in terms of 'a'.]

Context: Consider a set of numbers from the calendar (having endless rows) forming under the following shape:

We see that the total sum is always 5 times the number in the centre.

Q. Find other shapes for which the sum of the numbers within the figure is always a multiple of one of the numbers.

How many matchsticks will there be in Step 33, Step 84, and Step 108? Of course, we can draw and count, but is there a quicker way to find the answers using the pattern present here?

The number of matchsticks needed to make 33 triangles (Step 33) is ________. Similarly, find the number of matchsticks needed for Step 84 and Step 108.

Context: The expressions describing the rule/formula to find the number of matchsticks at step $y$ are $3 + 2 \times (y - 1)$ and $2y + 1$.

Q. Does the above expression also give the number of matchsticks at each step correctly? Are these expressions the same?

Context: Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle. For example, in step 2 there are 2 matchsticks placed horizontally and 3 matchsticks placed diagonally.

Q. What are these numbers in Step 3 and Step 4?

Context: Matchsticks are placed in two orientations — (a) horizontal ones at the top and bottom, and (b) the ones placed diagonally in the middle. For example, in step 2 there are 2 matchsticks placed horizontally and 3 matchsticks placed diagonally.

Q. How does the number of matchsticks change in each orientation as the steps increase? Write an expression for the number of matchsticks at Step ‘y’ in each orientation. Do the two expressions add up to 2y + 1?

15. Numbers are written in a particular sequence in this endless 4-column grid.

(a) Give expressions to generate all the numbers in a given column (1, 2, 3, 4).

(b) In which row and column will the following numbers appear:

(i) 124

(ii) 147

(iii) 201

(c) What number appears in row $r$ and column $c$?

(d) Observe the positions of multiples of 3.

Do you see any pattern in it? List other patterns that you see.